Abstract

Under the hypothesis that the
first derivative satisfies some kind of weak Lipschitz conditions, a new
semilocal convergence theorem for inexact Newton method is presented.
Unified convergence criteria ensuring the convergence of inexact Newton method
are also established. Applications to some special cases such as the Kantorovich
type conditions and 𝛾-Conditions are provided and some well-known
convergence theorems for Newton's method are obtained as corollaries.

1. Introduction

Let 𝐹 be a continuously Fréchet differentiable nonlinear operator from a convex subset 𝐷 of Banach space 𝑋 to Banach space 𝑌. Finding solutions of a nonlinear operator equation:
𝐹(𝑥)=0(1.1)
in Banach space is a basic and important problem in applied and computational mathematics. A classical method for finding an approximation of a solution of (1.1) is Newton's method which is defined by
𝑥𝑛+1=𝑥𝑛−𝐹𝑥𝑛−1𝐹𝑥𝑛,𝑥0∈𝐷,𝑛=0,1,2,….(1.2)

There is a huge literature on local as well as semilocal convergence for Newton's method under various assumptions (see [1–9]). Besides, there are a lot of works on the weakness of the hypotheses made on the underlying operators, see for example [2, 3, 5–9] and references therein. In particular, Wang in [7, 8] introduced the notions of Lipschitz conditions with 𝐿 average, under which Kantorovich like convergence criteria and Smale's point estimate theory can be put together to be investigated.

However, Newton's method has two disadvantages. One is to evaluate 𝐹 involved, the other is to solve the exact solution of Newton equations:
𝐹𝑥𝑛𝑥𝑛+1−𝑥𝑛𝑥=−𝐹𝑛,𝑛=0,1,2,….(1.3)
In many applications, for example, those in Euclidean spaces, computing the exact solutions using a direct method such as Gaussian elimination can be expensive if the number of unknowns is large and may not be justified when 𝑥𝑘 is far from the searched solution. While using linear iterative methods to approximate the solutions of (1.3) instead of solving it exactly can reduce some of the costs of Newton's method. One of the methods is inexact Newton method which can be found in [10] and takes the following form:
𝑥𝑛+1=𝑥𝑛+𝑠𝑛,𝐹𝑥𝑛𝑠𝑛𝑥=−𝐹𝑛+𝑟𝑛,𝑛=0,1,2,…,(1.4)
where {𝑟𝑛} is a sequence in 𝑌.

As is well known, the convergence behavior of the inexact Newton method depends on the residual controls of {𝑟𝑛} under the hypothesis that 𝐹′ satisfies different conditions. Some relative results can be found in [10–24], for example.

Under the Lipschitz continuity assumption on 𝐹′, different residual controls were used. For example, the residual controls ‖𝑟𝑛‖≤𝜂𝑛‖𝐹(𝑥𝑛)‖ were adopted in [10, 12]; in [15] the affine invariant conditions ‖𝐹′(𝑥0)−1𝑟𝑛‖≤𝜂𝑛‖𝐹′(𝑥0)−1𝐹(𝑥𝑛)‖ were considered; while in [21] Shen has analyzed the semilocal convergence behavior in some manner such that the relative residuals {𝑟𝑛} satisfy
‖‖𝑥𝐹′0−1𝑟𝑛‖‖≤𝜂𝑛‖‖𝐹𝑥0−1𝐹𝑥𝑛‖‖1+𝜅,0≤𝜅≤1,𝑛=0,1,2,….(1.5)
Assuming that the residuals satisfy ‖𝑃𝑛𝑟𝑛‖≤𝜃𝑛‖𝑃𝑛𝐹(𝑥𝑛)‖1+𝜅, where {𝑃𝑛} is a sequence of invertible operators from 𝑌 to 𝑋, and that 𝐹′(𝑥0)−1𝐹′ satisfies the Hölder condition around 𝑥0, Li and Shen established the local and semilocal convergence in [16, 20], respectively. Besides, the 𝛾-condition was also introduced into inexact Newton method in [22] by considering residual controls (1.5) with 𝜅=1, that is,
‖‖𝑥𝐹′0−1𝑟𝑛‖‖≤𝜂𝑛‖‖𝐹𝑥0−1𝐹𝑥𝑛‖‖2,𝑛=0,1,2,…;(1.6)
Smale's 𝛼-theory for the inexact Newton method was established there.

In the present paper, by considering the residual controls (1.6), we will study the convergence of inexact Newton method under the assumption that 𝐹 has a continuous derivative in a closed ball 𝐵(𝑥0,𝑟), 𝐹′(𝑥0)−1𝐹′ exists and 𝐹′(𝑥0)−1𝐹′ satisfies the weak Lipschitz condition:
‖‖𝐹𝑥0−1𝐹(𝑥)−𝐹𝑥‖‖≤𝜌(𝑥𝑥′)𝜌(𝑥)𝑥𝐿(𝑢)𝑑𝑢,∀𝑥∈𝐵0,𝑟,∀𝑥′∈𝐵(𝑥,𝑟−𝜌(𝑥)),(1.7)
where 𝑟 is a positive number, 𝜌(𝑥)=‖𝑥−𝑥0‖,𝜌(𝑥𝑥)=𝜌(𝑥)+‖𝑥−𝑥‖≤𝑟, and 𝐿 is a positive integrable nondecreasing function on [0,𝑟]. We also establish the unified convergence criteria, which include Kantorovich type and Smale type convergence criteria as special cases. In particular, in the special case when 𝜂𝑛=0(𝑛=0,1,2,…), (1.4) reduces to Newton's method and our result extends the corresponding one in [7].

The paper is organized as follows. Section 2 gives some lemmas which are used in the proof of our main theorem. In Section 3, the semilocal convergence of inexact Newton method is studied under the weak Lipschitz condition (1.7). Its applications to some special cases are provided in Section 4.

2. Preliminaries

Let 𝑋 and 𝑌 be Banach spaces. Throughout this paper, 𝑅>𝑟 are two positive numbers, 𝐿 is a positive integrable nondecreasing function on any involved intervals, and 𝐵(𝑥,𝑅) is an open ball in 𝑋 with center 𝑥 and radius 𝑅. Let 𝛽>0,0≤𝜆<1,𝜔≥1, and 𝜎≥0. Define
𝜑(𝑡)=𝛽−(1−𝜆)𝑡+𝜎𝑡2+𝜔𝑡0𝐿(𝑢)(𝑡−𝑢)𝑑𝑢,0≤𝑡≤𝑅,𝜓(𝑡)=𝛽−𝑡+𝜔𝑡0𝐿(𝑢)(𝑡−𝑢)𝑑𝑢,0≤𝑡≤𝑅.(2.1)
Obviously,
𝜑(𝑡)=−(1−𝜆)+2𝜎𝑡+𝜔𝑡0𝜓𝐿(𝑢)𝑑𝑢,0≤𝑡≤𝑅,(2.2)(𝑡)=−1+𝜔𝑡0𝜑𝐿(𝑢)𝑑𝑢,0≤𝑡≤𝑅,(2.3)(𝑡)=2𝜎+𝜔𝐿(𝑡)>0,0≤𝑡≤𝑅.(2.4)
Set
𝑟𝜆∶=sup𝑟∈(0,𝑅)∶𝜔𝑟0𝑏𝐿(𝑢)𝑑𝑢+2𝜎𝑟≤1−𝜆,(2.5)𝜆∶=(1−𝜆)𝑟𝜆−𝜎𝑟2𝜆−𝜔𝑟𝜆0𝑟𝐿(𝑢)𝜆−𝑢𝑑𝑢.(2.6)
Write ∫𝛿=𝜔𝑅0𝐿(𝑢)𝑑𝑢+2𝜎𝑅. Then
𝑟𝜆=𝑟𝑅,if𝛿<1−𝜆,𝜆,if𝛿≥1−𝜆,(2.7)
where 𝑟𝜆∈[0,𝑅] is such that 𝜔∫𝑟′𝜆0𝐿(𝑢)𝑑𝑢+2𝜎𝑟𝜆=1−𝜆. Furthermore, it follows that
𝑏𝜆≥𝜔𝑟𝜆0𝐿(𝑢)𝑢𝑑𝑢+𝜎𝑟2𝜆𝑏,if𝛿<1−𝜆,𝜆=𝜔𝑟𝜆0𝐿(𝑢)𝑢𝑑𝑢+𝜎𝑟2𝜆,if𝛿≥1−𝜆.(2.8)
Let
𝑡0=0,𝑡𝑛+1=𝑡𝑛−𝜑𝑡𝑛𝜓𝑡𝑛,𝑛=0,1,2,….(2.9)

The following two lemmas describe some properties about the majorizing function 𝜑 and the convergence property of {𝑡𝑛}.

Lemma 2.1. Suppose that 𝛽≤𝑏𝜆 and 𝜑 is defined by (2.1). Then the function 𝜑 is strictly decreasing and has exact one zero 𝑡∗ on [0,𝑟𝜆] satisfying 𝛽<𝑡∗.

Proof. By (2.4) and (2.5), we know 𝜑′ is strictly increasing on [0,𝑟𝜆] and has the values 𝜑′(0)<0 and 𝜑′(𝑟𝜆)≤0. This implies that 𝜑 is strictly decreasing on [0,𝑟𝜆]. Note that 𝜑(0)=𝛽>0 and 𝜑(𝑟𝜆)≤0 by the definition of 𝑏𝜆. Thus, 𝜑(𝑡)=0 has exact one solution 𝑡∗ on [0,𝑟𝜆]. Since
𝜑(𝛽)=𝜆𝛽+𝜎𝛽2+𝜔𝛽0𝐿(𝑢)(𝛽−𝑢)𝑑𝑢>0,(2.10)
we have 𝛽<𝑡∗. The proof is complete.

Lemma 2.2. Let 𝑡∗ be the positive solution of equation 𝜑(𝑡)=0 on [0,𝑟𝜆]. Suppose that 𝛽≤𝑏𝜆 and the sequence {𝑡𝑛} is defined by (2.9). Then
𝑡𝑛<𝑡𝑛+1<𝑡∗,𝑛=0,1,2,….(2.11)
Consequently, {𝑡𝑛} is strictly increasing and converges to 𝑡∗.

Proof. We prove the lemma by mathematical induction. Note that 0=𝑡0<𝑡1=𝛽<𝑡∗. For 𝑛>1, assume that
𝑡𝑛−1<𝑡𝑛<𝑡∗.(2.12)
Since 𝜓(𝑡)=𝜔𝐿(𝑡)>0,−𝜓 is strictly decreasing on [0,𝑟𝜆]. Hence,
−𝜓𝑡𝑛>−𝜓𝑡∗≥−𝜓𝑟𝜆=−𝜑𝑟𝜆+𝜆+2𝜎𝑟𝜆≥0.(2.13)
Moreover, 𝜑(𝑡𝑛)>0 by of Lemma 2.1. It follows that
𝑡𝑛+1=𝑡𝑛−𝜑𝑡𝑛𝜓𝑡𝑛>𝑡𝑛.(2.14)
Define a function 𝑁(𝑡) on [0,𝑡∗] by
𝑁(𝑡)∶=𝑡−𝜑(𝑡)𝜓(𝑡),𝑡∈0,𝑡∗.(2.15)
Note that 𝜓′(𝑡)<0,𝑡∈[0,𝑡∗], unless 𝜆=0, 𝜎=0 and 𝑡=𝑡∗=𝑟𝜆, for which we adopt the convention that lim𝑡→𝑡∗(𝜑(𝑡)/𝜓(𝑡))=0 and 𝑁(𝑡∗)=𝑡∗−lim𝑡→𝑡∗(𝜑(𝑡)/𝜓′(𝑡))=𝑡∗. Hence, the function 𝑁(𝑡) is well defined and continuous on [0,𝑡∗].Moreover, by (2.2) and (2.3), we have
𝑁𝜑(𝑡)=1−(𝑡)𝜓(𝑡)−𝜑(𝑡)𝜓(𝑡)(𝜓(𝑡))2=−𝜓(𝑡)(𝜆+2𝜎𝑡)+𝜑(𝑡)𝜓(𝑡)(𝜓(𝑡))2>0,𝑡∈0,𝑡∗.(2.16)
Hence, 𝑁(𝑡) is monotonically increasing on [0,𝑡∗). This together with (2.9) and (2.14) implies that
𝑡𝑛<𝑡𝑛+1𝑡=𝑁𝑛𝑡<𝑁∗=𝑡∗.(2.17)
Therefore, by mathematical induction, (2.11) holds. Consequently, {𝑡𝑛} is increasing, bounded, and converges to a point 𝑡∗𝜆, which satisfies 𝜑(𝑡∗𝜆)=0. Hence, 𝑡∗=𝑡∗𝜆. The proof is complete.

To prove our main result, we need two more lemmas. The first can be found in [23] and the second in [7].

Lemma 2.3. Suppose that 𝐹 has a continuous derivative satisfying the weak Lipschitz condition (1.7). Let 𝑟 satisfy ∫𝑟0𝐿(𝑢)𝑑𝑢≤1. Then 𝐹′(𝑥) is invertible in the ball 𝐵(𝑥0,𝑟) and
‖‖𝐹(𝑥)−1𝐹𝑥0‖‖≤1−0𝜌(𝑥)𝐿(𝑢)𝑑𝑢−1.(2.18)

3. Semilocal Convergence Analysis

Recall that 𝐹∶𝐷⊆𝑋→𝑌 is a nonlinear operator with continuous Fréchet derivative. Let 𝐵(𝑥0,𝑅)⊆𝐷 and 𝑥0∈𝐷 be such that 𝐹′(𝑥0)−1 exists. In the present paper, we adopt the residuals {𝑟𝑛} satisfying (1.6) and assume that 𝜂=sup𝑛≥0𝜂𝑛<1. Thus, if 𝑛≥0 and {𝑥𝑛} is well defined, then
‖‖𝑥𝐹′0−1𝑟𝑛‖‖≤𝜂𝑛‖‖𝐹𝑥0−1𝐹𝑥𝑛‖‖2‖‖𝐹≤𝜂𝑥0−1𝐹𝑥𝑛‖‖2.(3.1)
Let
‖‖𝐹𝛼=𝑥0−1𝐹𝑥0‖‖√,𝛽=1+𝜂𝛼.(3.2)
Write
√𝜔=1+𝜂√𝜂,𝜎=1+𝜂∫1+𝑅0𝐿(𝑢)𝑑𝑢2√1−𝜂2.(3.3)
Recall that 𝑟𝜆 is determined by (2.5), 𝜑(𝑡∗)=0, and {𝑡𝑛} is generated by (2.9) with 𝜔 and 𝜎 given in (3.3).

Lemma 3.1. Let {𝑥𝑛} be a sequence generated by (1.4). Suppose that F satisfies the weak Lipschitz condition (1.7) on 𝐵(𝑥0,𝑡∗)⊆𝐵(𝑥0,𝑅) and that 𝛽≤𝑏𝜆. For an integer 𝑚≥1, if
√𝜂‖‖𝐹𝑥0−1𝐹𝑥𝑛−1‖‖‖‖𝑥≤1,𝑛−𝑥𝑛−1‖‖≤𝑡𝑛−𝑡𝑛−1(3.4)
hold for each 1≤𝑛≤𝑚, then the following assertions hold:
√1+𝜂‖‖𝐹𝑥0−1𝐹𝑥𝑚‖‖𝑡≤𝜑𝑚;√𝜂‖‖𝐹𝑥0−1𝐹𝑥𝑚‖‖≤1.(3.5)

Proof. Assume that (3.4) holds for each 1≤𝑛≤𝑚. Write 𝑥𝜏𝑚−1=𝑥𝑚−1+𝜏(𝑥𝑚−𝑥𝑚−1),𝜏∈[0,1]. Applying (1.4), we have
𝐹𝑥𝑚𝑥=𝐹𝑚𝑥−𝐹𝑚−1−𝐹𝑥𝑚−1𝑥𝑚−𝑥𝑚−1+𝑟𝑚−1=10𝐹𝑥𝜏𝑚−1−𝐹𝑥𝑚−1𝑥𝑑𝜏𝑚−𝑥𝑚−1+𝑟𝑚−1.(3.6)
Hence,
‖‖𝐹𝑥0−1𝐹𝑥𝑚‖‖≤‖‖‖𝑥𝐹′0−110𝐹𝑥𝜏𝑚−1−𝐹𝑥𝑚−1𝑥𝑑𝜏𝑚−𝑥𝑚−1‖‖‖+‖‖𝑥𝐹′0−1𝑟𝑚−1‖‖=𝐼1+𝐼2.(3.7)
To estimate 𝐼1, by (3.4), we notice that
‖‖𝑥𝜏𝑚−1−𝑥0‖‖=‖‖𝑥𝑚−1𝑥+𝜏𝑚−𝑥𝑚−1−𝑥0‖‖≤𝑚−1𝑛=1‖‖𝑥𝑛−𝑥𝑛−1‖‖‖‖𝑥+𝜏𝑚−𝑥𝑚−1‖‖≤𝑡𝑚−1𝑡+𝜏𝑚−𝑡𝑚−1=(1−𝜏)𝑡𝑚−1+𝜏𝑡𝑚<𝑡∗.(3.8)
In particular,
‖‖𝑥𝑚−1−𝑥0‖‖≤𝑡𝑚−1<𝑡∗,‖‖𝑥𝑚−𝑥0‖‖≤𝑡𝑚<𝑡∗.(3.9)
Thus, by the weak Lipschitz condition (1.7), we obtain
𝐼1≤‖𝑥𝑚−𝑥𝑚−1‖0‖‖𝑥𝑚−𝑥𝑚−1‖‖𝐿‖‖𝑥−𝑢𝑚−1−𝑥0‖‖+𝑢𝑑𝑢.(3.10)
Below we estimate 𝐼2. We firstly notice that (3.1) and (3.4) yield
‖‖𝐹𝑥0−1𝐹𝑥𝑚−1𝑥𝑚−𝑥𝑚−1‖‖≥‖‖𝑥𝐹′0−1𝐹𝑥𝑚−1‖‖−‖‖𝑥𝐹′0−1𝑟𝑚−1‖‖≥‖‖𝐹𝑥0−1𝐹𝑥𝑚−1‖‖‖‖𝑥−𝜂𝐹′0−1𝐹𝑥𝑚−1‖‖2≥√1−𝜂‖‖𝐹𝑥0−1𝐹𝑥𝑚−1‖‖.(3.11)
Since
‖‖𝐹𝑥0−1𝐹𝑥𝑚−1‖‖=‖‖𝐼+𝐹𝑥0−1𝐹𝑥𝑚−1−𝐹𝑥0‖‖≤1+𝜌(𝑥𝑚−1)0𝐿(𝑢)𝑑𝑢≤1+𝑅0𝐿(𝑢)𝑑𝑢,(3.12)
we have
‖‖𝐹𝑥0−1𝐹𝑥𝑚−1‖‖≤‖‖𝐹𝑥0−1𝐹𝑥𝑚−1‖‖‖‖𝑥𝑚−𝑥𝑚−1‖‖√1−𝜂≤∫1+𝑅0𝐿(𝑢)𝑑𝑢√1−𝜂‖‖𝑥𝑚−𝑥𝑚−1‖‖.(3.13)
Combining this with (3.1) implies that
𝐼2‖‖𝐹≤𝜂𝑥0−1𝐹𝑥𝑚−1‖‖2≤𝜂∫1+𝑅0𝐿(𝑢)𝑑𝑢2√1−𝜂2‖‖𝑥𝑚−𝑥𝑚−1‖‖2.(3.14)
Consequently, by (3.7), (3.10), (3.14) and Lemma 2.4, we get
√1+𝜂‖‖𝐹𝑥0−1𝐹𝑥𝑚‖‖≤√1+𝜂𝐼1+𝐼2≤√1+𝜂‖𝑥𝑚−𝑥𝑚−1‖0‖‖𝑥𝑚−𝑥𝑚−1‖‖𝐿‖‖𝑥−𝑢𝑚−1−𝑥0‖‖+𝜂√+𝑢𝑑𝑢1+𝜂∫1+𝑅0𝐿(𝑢)𝑑𝑢2√1−𝜂2‖‖𝑥𝑚−𝑥𝑚−1‖‖2=𝜔‖𝑥𝑚−𝑥𝑚−1‖0‖‖𝑥𝑚−𝑥𝑚−1‖‖𝐿‖‖𝑥−𝑢𝑚−1−𝑥0‖‖‖‖𝑥+𝑢𝑑𝑢+𝜎𝑚−𝑥𝑚−1‖‖2=𝜔‖‖𝑥𝑚−𝑥𝑚−1‖‖2‖𝑥𝑚−𝑥𝑚−1‖0‖‖𝑥𝑚−𝑥𝑚−1‖‖‖‖𝑥−𝑢×𝐿𝑚−1−𝑥0‖‖‖‖𝑥+𝑢𝑑𝑢+𝜎𝑚−𝑥𝑚−1‖‖2≤𝜔𝑡𝑚−𝑡𝑚−12𝑡𝑚−𝑡𝑚−10𝑡𝑚−𝑡𝑚−1𝐿𝑡−𝑢𝑚−1×𝑡+𝑢𝑑𝑢+𝜎𝑚−𝑡𝑚−12=𝜔𝑡𝑚−𝑡𝑚−10𝑡𝑚−𝑡𝑚−1𝐿𝑡−𝑢𝑚−1𝑡+𝑢𝑑𝑢+𝜎2𝑚−𝑡2𝑚−1−2𝑡𝑚−1𝑡𝑚−𝑡𝑚−1𝑡=𝜑𝑚𝑡−𝜑𝑚−1−𝜑𝑡𝑚−1𝑡𝑚−𝑡𝑚−1.(3.15)
Noting that 𝜑′(𝑡)=𝜓′(𝑡)+𝜆+2𝜎𝑡 and −𝜑(𝑡𝑚−1)−𝜓′(𝑡𝑚−1)(𝑡𝑚−𝑡𝑚−1)=0, we have
√1+𝜂‖‖𝐹𝑥0−1𝐹𝑥𝑚‖‖𝑡≤𝜑𝑚𝑡−𝜑𝑚−1−𝜑𝑡𝑚−1𝑡𝑚−𝑡𝑚−1𝑡=𝜑𝑚−𝜆+2𝜎𝑡𝑚−1𝑡𝑚−𝑡𝑚−1𝑡≤𝜑𝑚.(3.16)
Moreover, since 𝜑 is decreasing on [0, 𝑡∗], one has
√1+𝜂‖‖𝐹𝑥0−1𝐹𝑥𝑚‖‖𝑡≤𝜑𝑚𝑡≤𝜑0=𝛽.(3.17)
And therefore
√𝜂‖‖𝐹𝑥0−1𝐹𝑥𝑚‖‖≤√𝜂√1+𝜂√𝛽=𝜂‖‖𝐹𝑥0−1𝐹𝑥0‖‖≤1.(3.18)
That is, (3.5) holds, and the proof is complete.

We now give the main result.

Theorem 3.2. Suppose that √𝛽≤min{1/𝜂,𝑏𝜆} and 𝐵(𝑥0,𝑡∗)⊆𝐵(𝑥0,𝑅), and that 𝐹′(𝑥0)−1𝐹′ satisfies the weak Lipschitz condition (1.7) on 𝐵(𝑥0,𝑡∗). Then the sequence {𝑥𝑛} generated by the inexact Newton method (1.4) converges to a solution 𝑥∗ of (1.1). Moreover,
‖‖𝑥𝑛−𝑥∗‖‖≤𝑡∗−𝑡𝑛,𝑛=0,1,2,….(3.19)

Proof. We firstly use mathematical induction to prove that (3.4) holds for each 𝑛=1,2,…. For 𝑛=1, by the above condition and (3.2), the first inequality in (3.4) holds trivially. While the second one can be proved as follows:
‖‖𝑥1−𝑥0‖‖≤‖‖𝐹𝑥0−1𝐹𝑥0‖‖+‖‖𝑥𝐹′0−1𝑟0‖‖≤𝛼+𝜂𝛼2√≤𝛼+√𝜂𝛼=1+𝜂𝛼=𝛽=𝑡1−𝑡0.(3.20)
Assume that (3.4) holds for all 𝑛≤𝑚. Then, Lemma 3.1 is applicable to concluding that
√1+𝜂‖‖𝐹𝑥0−1𝐹𝑥𝑚‖‖𝑡≤𝜑𝑚;√𝜂‖‖𝐹𝑥0−1𝐹𝑥𝑚‖‖≤1.(3.21)
Hence, by (3.5), together with the weak Lipschitz condition (1.7) and Lemma 2.3, one has
‖‖𝑥𝑚+1−𝑥𝑚‖‖≤‖‖𝐹𝑥𝑚−1𝐹𝑥0‖‖‖‖𝐹𝑥0−1𝐹𝑥𝑚‖‖+‖‖𝑥𝐹′0−1𝑟𝑚‖‖≤1∫1−𝜌(𝑥𝑚)0‖‖𝐹𝐿(𝑢)𝑑𝑢𝑥0−1𝐹𝑥𝑚‖‖‖‖+𝜂𝐹′(𝑥𝑚)−1𝐹(𝑥𝑚)‖‖2≤√1+𝜂∫1−𝜔𝜌(𝑥𝑚)0‖‖𝐹𝐿(𝑢)𝑑𝑢𝑥0−1𝐹𝑥𝑚‖‖𝜑𝑡≤−𝑚𝜓𝑡𝑚=𝑡𝑚+1−𝑡𝑚.(3.22)
Therefore, (3.4) holds for 𝑛=𝑚+1 and so for each 𝑛≥1. Consequently, for 𝑛≥0 and 𝑘≥0,
‖‖𝑥𝑘+𝑛−𝑥𝑛‖‖≤𝑘𝑖=1‖‖𝑥𝑖+𝑛−𝑥𝑖+𝑛−1‖‖≤𝑘𝑖=1𝑡𝑖+𝑛−𝑡𝑖+𝑛−1=𝑡𝑘+𝑛−𝑡𝑛.(3.23)
This together with Lemma 2.2 means that {𝑥𝑛} is a Cauchy sequence and so converges to some 𝑥∗. While taking 𝑘→∞ in (3.23), we obtain
‖‖𝑥𝑛−𝑥∗‖‖≤𝑡∗−𝑡𝑛,𝑛=0,1,2,….(3.24)
The proof is complete.

Corollary 3.3. Assume that 𝛽≤𝑏𝜆 and 𝐵(𝑥0,𝑡∗)⊆𝐵(𝑥0,𝑅), where 𝑏𝜆=∫𝑟𝜆0𝐿(𝑢)𝑢𝑑𝑢 and 𝑟𝜆 satisfying ∫𝑟𝜆0𝐿(𝑢)𝑑𝑢≤1−𝜆. Suppose that 𝐹′(𝑥0)−1𝐹′ satisfies the weak Lipschitz condition (1.7) on 𝐵(𝑥0,𝑡∗). Then the sequence {𝑥𝑛} generated by Newton's method (1.2) converges to a solution 𝑥∗ of (1.1). Moreover,
‖‖𝑥𝑛−𝑥∗‖‖≤𝑡∗−𝑡𝑛,𝑛=0,1,2,…,(3.25)
where 𝑡∗ and {𝑡𝑛} are defined in Lemma 2.2 for 𝜂=0.

In more particular, suppose that ∫𝑅0𝐿(𝑢)𝑑𝑢>1 and 𝜆=0. Then Corollary 3.3 reduces to the following result given in (Theorem 3.1, [7]).

Corollary 3.4. Assume that 𝛽≤𝑏𝜆0, where 𝑏𝜆0=∫𝑟𝜆00𝐿(𝑢)𝑢𝑑𝑢 and ∫𝑟𝜆00𝐿(𝑢)𝑑𝑢=1. Suppose that 𝐹′(𝑥0)−1𝐹′ satisfies weak Lipschitz condition (1.7) on 𝐵(𝑥0,𝑡∗)⊆𝐵(𝑥0,𝑅). Then the sequence {𝑥𝑛} generated by Newton's method (1.2) converges to a solution 𝑥∗ of (1.1). Moreover,
‖‖𝑥𝑛−𝑥∗‖‖≤𝑡∗−𝑡𝑛,𝑛=0,1,2,…,(3.26)
where 𝑡∗ and {𝑡𝑛} are defined in Lemma 2.2 for 𝜂=0 and 𝜆=0.

4. Application

This section is divided into two subsections: we consider the applications of our main results specializing, respectively, in Kantorovich type condition and in 𝛾-condition. In particular, our results reduce some of the corresponding results of Newton's method.

4.1. Kantorovich-Type Condition

Throughout this subsection, let 𝐿 be a positive constant. By (2.1), we have
1𝜑(𝑡)=𝛽−(1−𝜆)𝑡+𝜎+2𝑡𝜔𝐿21,𝑡≥0,𝜓(𝑡)=𝛽−𝑡+2𝜔𝐿𝑡2,𝑡≥0.(4.1)
By (2.5) and (2.6), we get
𝑟𝜆=1−𝜆𝜔𝐿+𝜎,𝑏𝜆=(1−𝜆)2𝜔𝐿2.(𝜔𝐿+𝜎)(4.2)
The convergence criterion becomes
‖‖𝐹𝑥0−1𝐹𝑥0‖‖≤(1−𝜆)2𝜔𝐿2.(𝜔𝐿+𝜎)(4.3)

Moreover, suppose that 𝜂=0 and 𝜆=0. Then criterion (4.3) reduces to the well-known Kantorovich type criterion ‖𝐹′(𝑥0)−1𝐹(𝑥0)‖≤1/2𝐿 of Newton's method in [7].

Corollary 4.1. Let 𝐿 be a positive constant, 𝛽=‖𝐹′(𝑥0)−1𝐹(𝑥0)‖ and 𝛽≤𝑏𝜆0, where 𝑏𝜆0=1/2𝐿 and 𝑟𝜆0=1/𝐿. Assume that 𝐹 satisfies the condition:
‖‖𝐹𝑥0−1𝐹(𝑥)−𝐹𝑥‖‖‖‖≤𝐿𝑥−𝑥‖‖,∀𝑥,𝑥𝑥∈𝐵0,‖‖,𝑟𝑥−𝑥0‖‖+‖𝑥−𝑥′‖≤𝑟,(4.4)
where √𝑟=(1−1−2𝐿𝛽)/𝐿. Then the sequence {𝑥𝑛} generated by Newton's method (1.2) converges to a solution 𝑥∗ of (1.1), and satisfies
‖‖𝑥𝑛−𝑥∗‖‖≤𝑡∗−𝑡𝑛,𝑛=0,1,2,….(4.5)

4.2. 𝛾-Condition

Throughout this subsection, we assume that 𝛾>0 and 𝐹 has continuous second derivative and satisfies
‖‖𝐹𝑥0−1𝐹‖‖≤(𝑥)2𝛾‖‖1−𝛾𝑥−𝑥0‖‖3𝑥,∀𝑥∈𝐵0,1𝛾.(4.6)
Let
𝐿(𝑢)=2𝛾(1−𝛾𝑢)31,𝑢∈0,𝛾.(4.7)
Then, by (2.1), we have
𝜑(𝑡)=𝛽−(1−𝜆)𝑡+𝜎𝑡2+𝛾𝑡211−𝛾𝑡,0≤𝑡<𝛾,𝜓(𝑡)=𝛽−𝑡+𝛾𝑡211−𝛾𝑡,0≤𝑡<𝛾.(4.8)
By (2.5) and (2.6), 𝑟𝜆 and 𝑏𝜆 satisfy
𝜔11−𝛾𝑟𝜆2−1+𝜎𝑟𝜆=1−𝜆,𝑏𝜆=𝛾𝑟2𝜆1−𝛾𝑟𝜆2.(4.9)
The convergence criterion becomes
‖‖𝐹𝑥0−1𝐹𝑥0‖‖≤𝛾𝑟2𝜆1−𝛾𝑟𝜆2.(4.10)

In the more special case, when 𝜂=0 and 𝜆=0, we obtain the criterion ‖𝐹′(𝑥0)−1𝐹(𝑥0√)‖≤(3−22)/𝛾 the same with Newton's method in [7].

Corollary 4.2. Let 𝛾 be a positive constant, 𝛽=‖𝐹′(𝑥0)−1𝐹(𝑥0)‖ and 𝛽≤𝑏𝜆0, where 𝑏𝜆0√=(3−22)/𝛾 and 𝑟𝜆0√=(1−(1/2))(1/𝛾). Assume that F satisfies the condition:
‖‖𝐹𝑥0−1𝐹(𝑥)−𝐹𝑥‖‖≤1‖‖1−𝛾𝑥−𝑥0‖‖‖‖𝑥−𝛾−𝑥0‖‖2−11−𝛾‖𝑥−𝑥0‖2,∀𝑥,𝑥𝑥∈𝐵0,‖‖,𝑟𝑥−𝑥0‖‖+‖𝑥′−𝑥‖≤𝑟,(4.11)
where √𝑟=(1+𝛽𝛾−(1+𝛽𝛾)2−8𝛽𝛾)/4𝛾. Then the sequence {𝑥𝑛} generated by Newton's method (1.2) converges to a solution 𝑥∗ of (1.1), and satisfies
‖‖𝑥𝑛−𝑥∗‖‖≤𝑡∗−𝑡𝑛,𝑛=0,1,2,….(4.12)

Acknowledgment

Supported in part by the National Natural Science Foundation of China (Grants no. 61170109 and no. 10971194) and Zhejiang Innovation Project (Grant no. T200905).